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# If n is an integer, which of the following must also be an integer?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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If n is an integer, which of the following must also be an integer?  [#permalink]

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18 Jul 2018, 05:04
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45% (medium)

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63% (01:13) correct 37% (01:19) wrong based on 118 sessions

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[GMAT math practice question]

If n is an integer, which of the following must also be an integer?

I. $$\frac{n(n+1)}{2}$$
II. $$\frac{n(n+1)(n+2)}{6}$$
III. $$\frac{n(n+1)(n+2)(n+3)}{8}$$

A. I only
B. II only
C. III only
D. I and II
E. I, II and III

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" RC Moderator Status: Perfecting myself for GMAT Joined: 22 May 2017 Posts: 433 Concentration: Nonprofit Schools: Haas '21 GPA: 4 WE: Engineering (Computer Software) If n is an integer, which of the following must also be an integer? [#permalink] ### Show Tags 18 Jul 2018, 07:45 1 I) Since n is an integer, either n or n+1 has to be even. S0, one of n and (n+1) is divisible by 2 $$\frac{n(n+1)}{2}$$ is an integer II) The product of any 3 consecutive integers is divisible by 3 => one of n, n+1, n+2 is divisible by 3 one of n, n+1, n+2 is also even. So, $$n(n+1)(n+2)$$ is divisible by both 3 and 2 and hence is divisible by 6 $$\frac{n(n+1)(n+2)}{6}$$ yields an integer III) $$n(n+1)(n+2)(n+3)$$ The product of any 4 consecutive integers includes a multiple of 2 and a multiple of 4 provided none of the 4 integers is 0 => $$n(n+1)(n+2)(n+3)$$ is divisible by 8 => $$\frac{n(n+1)(n+2)(n+3)}{8}$$ yields an integer Hence option E _________________ If you like my post press kudos +1 New - RC Butler - 2 RC's everyday Tag me in RC questions if you need help. Please provide your analysis of the question in the post along with the tag. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6028 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If n is an integer, which of the following must also be an integer? [#permalink] ### Show Tags 20 Jul 2018, 01:28 => Statement I Since $$n$$ and $$n + 1$$ are consecutive integers, $$n(n+1)$$ is a multiple of $$2$$. Thus, $$\frac{n(n+1)}{2}$$ is an integer. Statement II Since $$n$$ and $$n + 1$$ are consecutive integers, $$n(n+1)$$ and $$n(n+1)(n+2)$$ are multiples of $$2$$. Since $$n, n + 1$$ and $$n + 2$$ are three consecutive integers, $$n(n+1)(n+2)$$ is a multiple of $$3$$. Thus, $$n(n+1)(n+2)$$ is a multiple of $$6$$, and $$\frac{n(n+1)(n+2)}{6}$$ is an integer. Statement III Since $$n$$ and $$n + 1$$ are two consecutive integers, $$n(n+1)$$ is a multiple of $$2$$. Similarly, $$(n+2)(n+3)$$ is a multiple of $$2$$. Also, either n and $$n + 2$$ or $$n + 1$$ and $$n + 3$$ are consecutive even integers. Thus, either $$(n + 1)(n+3)$$ is a multiple of $$8$$ or $$n(n+2)$$ is a multiple of $$8$$ since one of them is a multiple of $$4$$. It follows that $$n(n+1)(n+2)(n+3)$$ is a multiple of $$8$$, and $$\frac{n(n+1)(n+2)(n+3)}{8}$$ is an integer. Therefore, the answer is E. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If n is an integer, which of the following must also be an integer?  [#permalink]

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20 Jul 2018, 01:54
1
MathRevolution wrote:
[GMAT math practice question]

If n is an integer, which of the following must also be an integer?

I. $$\frac{n(n+1)}{2}$$
II. $$\frac{n(n+1)(n+2)}{6}$$
III. $$\frac{n(n+1)(n+2)(n+3)}{8}$$

A. I only
B. II only
C. III only
D. I and II
E. I, II and III

From1: n,n+1 is product of two consecutive integer so divisible by 2 :sufficient
From2: n,n+1,n+2 is product of three consecutive integer so divisible by 2 and 3 :sufficient
From1: n,n+1,n+2,n+3 is product of four consecutive integer so divisible by 8
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Re: If n is an integer, which of the following must also be an integer?  [#permalink]

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24 Jul 2018, 12:09
1
Quote:
If n is an integer, which of the following must also be an integer?

Is the question stem supposed to say "If n is a positive integer?"
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Joined: 04 Jun 2018
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Re: If n is an integer, which of the following must also be an integer?  [#permalink]

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24 Jul 2018, 12:44
1
Quote:
If n is an integer, which of the following must also be an integer?

Is the question stem supposed to say "If n is a positive integer?"

I had the same thougt, if n= -1 then n+1 is 0 and thus 1*0 / 2 is not an integer.
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Joined: 16 Aug 2015
Posts: 6028
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If n is an integer, which of the following must also be an integer?  [#permalink]

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30 Jul 2018, 08:45
Quote:
If n is an integer, which of the following must also be an integer?

Is the question stem supposed to say "If n is a positive integer?"

We don't need the assumption n is a positive integer.
It is valid for 0 or negative integers too.
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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6028 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If n is an integer, which of the following must also be an integer? [#permalink] ### Show Tags 30 Jul 2018, 08:46 Arro44 wrote: blackshadow1357 wrote: Quote: If n is an integer, which of the following must also be an integer? Is the question stem supposed to say "If n is a positive integer?" I had the same thougt, if n= -1 then n+1 is 0 and thus 1*0 / 2 is not an integer. 1*0 / 2 = 0 is an integer. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If n is an integer, which of the following must also be an integer? &nbs [#permalink] 30 Jul 2018, 08:46
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